Conservation of energy in cosmology

W. Jim Jastrzebski, file 164-19.htm
Unofficial PhD program supervised by prof. Józef Namysłowski
The Institute of Theoretical Physics, University of Warsaw
Room 139, Hoża 69, 00-681 Warsaw, Poland
It has been shown that the hypothesis of strict conservation of energy predicts many observations with about one sigma accuracy. Except explaining the mechanism of gravitational pseudo force, done by Einstein in 1915, verifiable directly by its value of F = gm and the mechanism of cosmological redshift verifiable by the value of Hubble "constant" Ho = c / Rs where c is speed of light and Rs is the radius of 3-space verifiable also by the position of minimum of angular diameters of galaxies as function of their redshifts, the hypothesis predicts for Ho = 70 km/s/Mpc the value of acceleration of cosmic probes (as Pioneer 10 and 11) 7×10-10 m/s2, and the average density of universe 6×10-27 kg/m3. The volume of space becomes 1.6×1012 Mpc3, and therefore the total mass of universe 3×1053 kg. The predicted acceleration of apparent expansion of space becomes dH/dt = -Ho2/2, and the distances to quasars several orders of magnitude smaller than those predicted by the hypothesis of expanding space.
         The hypothesis of expanding space requires that energy is not conserved globally. We show that the assumption of strict conservation of energy makes the hypothesis of expanding space redundant since the former predicts many observations quantitatively with accuracy better than one sigma which strongly suggests that it might be a true hypothesis and consequently the space is not expanding.
         The following section is overly detailed to be understood also by those astronomers and astrophysicists who don't understand general relativity but understand that the creation of energy may not exist.

Step-by-step derivation of cosmological redshift in stationary space
         Consider homogeneous universe in a form of Einstein's 3-sphere of radius Rs (and therefore of volume 2Rs3) [1] filled with dust of density ρ (with a particle of dust representing a galaxy). Let energy conservation holds. Let photons move through this dust interacting with it only gravitationally via dynamical friction [2] [3] [4] [5] [6] losing gradually their energy to this dust. Qualitatively, to the observer at some distance from the light source this energy transfer from photons to dust manifests itself as a change in wavelength, simulating the tired light effect, first observed by Hubble and then worked on by Zwicky. The relativistic interpretation of this result allows the derivation of cosmological redshift, including discovery that Hubble's "constant" depends on the distance between the light emitting place in deep space and the observer [7]. Let's do some math to find out what happens quantitatively. It turns out that the Newtonian math suffices for this purpose.
         Let photons of energy Eo are radiated out from an arbitrarily placed center of coordinate system called "point zero" in the described above homogeneous space. When the radiated out photons leave an empty hole at point zero in otherwise homogeneous space the space becomes inhomogeneous. The Newtonian "gravitational force" shows up and starts pushing the dust away from the empty hole at point zero like an air bubble in a tank of water "pushes away" with its "gravitational force" the surrounding water. The space around point zero gains gravitational energy that given enough time, gets eventually converted into kinetic energy of movement of dust particles. This effect refers to the future though that we don't need to be concerned with since we describe only the present, namely the gravitational energies, The future state of universe can have no influence on the present and its energies.
         Let the gravitational energy of dust contained in the volume of a ball of dust of radius r be Ed(r). At distance r from point zero the gravitational force pushing the dust particles away is equal according to Newton's math, using Einstein's relativistic relation between mass and energy E=mc2
  Fparticle = ( G / c2 )[ Eo - Ed(r) ] mparticle / r2 (1)
where mparticle is the mass of a particle of dust universe (typically a galaxy).
         Integrating the mass over all dust particles in a spherical shell of dust of radius r and thickness dr we get mass of this shell of dust at distance r from point zero as
  mshell = 4 π ρ r2 dr (2)
and merging (1) with (2) the total force that is the source of gravitational energy of dust of this shell becomes
  Fshell(r) = ( 4πGρ / c2 ) [ Eo - Ed(r) ] dr (3)
and since Fshell(r) = - dEshell(r) / dr we have also
  - dEshell(r) / dr = ( 4πGρ / c2 ) [ Eo - Ed(r) ] dr (4)
         Integrating both sides of (4) over all spherical shells between point zero and r to get total gravitational energy of dust, and differentiating both sides with respect to r to get rid of the integral on the right side of this equation, we get
  - d2Ed(r) / dr2 = ( 4πGρ / c2 ) [ Eo - Ed(r) ] (5)
and substitutig Ed(r) = Eo - Eph(r) where Eph(r) is energy of photons at distance r from their source and introducing square of radius of stationary universe Rs2 = c2/4πGρ we get the final differential equation controlling the phenomenon as
  d2Eph(r) / dr2 = Eph(r) / Rs2 (6)
         Solving equation (6) with initial conditions Eph|r=0 = Eo and dEph/dr|r=0 = - Eo / Rs meaning selecting a solution that makes physical sense, we get
  Eph(r) / Eo = exp( - r / Rs ) (7)
         Since we know that in general relativity there is nothing else but the time dilation and the curvature of space as the media controlling gravitation we now switch from the Newtonian math to the relativistic interpretation of the above result, which is the time running slower at a distance from (any) observer according to relation
  dτ/dt = exp( - r / Rs ) (8)
where τ is proper time at point in deep space and t is coordinate time at observer. The effect might be called Hubble time dilation in honor of its discoverer, and as distinguished from the gravitational time dilation discovered by Einstein.
         After differentiating the above equation at r = 0 we get a relation between the Hubble time dilation in deep space (d2τ/dtdr) and the square root of curvature of space (1 / Rs ) as
  d2τ/dtdr + 1 / Rs = 0 (9)
that suggests the existence of a tensor named here tentatively Hμν or Hubble tensor, modifying the right side of Einstein's field equation such that Hμν + Rμν = 0 indicating that the spacetime is intrinsically flat as proposed by Narlikar and Arp [8] and required by the law of conservation of energy (allowing also to remove the cosmological constant Λ from Einstein's field equation - a note for mathematicians rather than grannies who might even not know what Einstein's field equation is nor they need to know to understand this derivation).
         It follows from equation (7) or (8) equivalently, that the redshift, produced by Hubble time dilation, is equal to
  Z = exp( r / Rs ) - 1 (10)
simulating the expansion of space, with the Hubble "constant" of this apparent "expansion" at r = 0
  Ho = c / Rs (11)
and after expanding the Hubble "constant", H(t), into Taylor series around t = 0 and neglecting the higher order terms, the "acceleration" of this apparent "expansion" equals
  dH / dt = - Ho2 / 2 (12)
which agrees within one standard deviation with 1998 observations by the Supernova Cosmology Project team [9]. The additional gain from all the above is that, "Einstein's biggest blunder of his life" [10], the infamous "cosmological constant" [11], a.k.a. "dark energy", falls out from Einstein's field equation.

Some numerical results
         Assuming Ho = 70 km/s/Mpc, as it is presently observed, and the same lower limit of dynamical friction value for space probes (since in the above derivation there was no dependence on the velocity of photons and so it refered to any velocity, also that of cosmic probes as Pioneers 10 and 11) the deceleration of cosmic probes comes out as 7×10-10 m/s2, the volume of space of our universe is 1.6×1012 Mpc3 making room for about 1012 galaxies. The average density of universe is 6×10-27 kg/m3, and its total mass is 3×1053 kg making the mass of average galaxy equal 3×1041 kg or 1011 solar masses. The apparent "acceleration" of space dH/dt = - Ho2/2 as predicted in 1985 by Einsteinian gravitation and observed in 1998 [9]. The numerical results have been never allowed to be published though in any scientific journal (not even in a popular science magazine as "Scientific American") so the astronomers for whom it might be an important news couldn't learn that the hypothesis of creation of the universe might be wrong. That this hypothesis not only contradicts Einstein's general relativity (replaced by Wheeler's cerationist version of general relativity based on an axiom of creation) but contradicts also quite unnecessarily the principle of conservation of energy since as it has been shown above, this principle delivers quite reasonable results, not only qualitatively but also quantitatively.

         The analysis of cosmological redshift can be carried out in just dozen simple steps using general relativity and the law of conservation of energy. The observed cosmological redshift can be attributed to the time dilation coupled to the curvature of space in the universe which remains stationary as predicted by Einstein in 1917. In addition to reproducing the measured properties of cosmological redshift, the approach outlined in this paper allows direct calculation of the important parameters of our universe such as the average density of space and the acceleration of its apparent expansion. While the formula for cosmological redshift, equation (10), can be derived directly from equation (7) obtained using Newtonian approximation, it is equation (8), which expresses the essential transition from Newtonian approach in which space and time are distinct, to a general relativistic spacetime.
         The additional gain from the above derivation is realization that redshift of quasars being proportional to the local curvature of space integrated over the whole distance between the quasar and the observer, as equation (9) suggests, doesn't need to reflect anymore the Hubble distance/redshift relation since the local curvatures of space don't need to be the same as the average, and the local curvature might be even many orders of magnitude greater than the agerage that reflects the average cosmological redshift.

         The author expresses his gratitude to Dr Halton Arp, Dr Helmut A. Abt, Dr Chris E. Adamson, Prof. John Baez, Prof. Tadeusz Balaban, Prof. Michal Chodorowski, Dr Tom Cohoe, Prof. Marek Demianski, Dr Marijke Van Gans, Prof. Roy J. Glauber, Dr Mike Guillen, Prof. Alan Guth, Dr Martin J. Hardcastle, Dr Franz Haymann, Dr Chris Hillman, Dr Marek Kalinowski, Dr Alan P. Lightman, Prof. Krzysztof Meissner, Prof. Jozef M. Namyslowski, Dr Bjarne G. Nielsen, Prof. Bohdan Paczynski, Janina Pisera, BS, Dr Ramon Prasad, Dr Frank E. Reed, Dr Anrzej Szechter, Prof. Henryk Szymczak, Jerzy Tarasiuk, MS, Dr Michael S. Turner, Dr Slava Turyshev, Prof. Clifford M. Will, Prof. Ned Wright, and anonumous referees from various scientific journals, for the time they have spent discussing with the author the subject of the paper and related issues. The most thanks goes to Prof. Michal Chodorowski of CAMK-PAN (Copernican Astronomical Center of Polish Accademy of Science), without whose friendly critique the ideas expressed in this paper might have never developed to a legible state.

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